nonlinear microstrain theories2006

Nonlinear microstrain theories Samuel Forest a, * , Rainer Sievert b a Ecole des Mines de Paris/CNRS, Centre des Mate ´ riaux/UMR 7633, BP 87, 91003 Evry, France b Federal Institute for Materials Research and Testing (BAM), Division V.2 Mechanical Behaviour of Materials, D-12200 Berlin, Germany Recei ved 16 Septe mber 2005; received in revised form 24 Febru ary 2006 Available online 11 May 2006 Communicated by P.R. Onck Abstract A hierarchy of higher order continua is presented that introduces additional degrees of freedom accounting for volume changes, rotation and straining of an underlying microstructure. An increase in the number of degrees of freedom represents a refine ment of the material descripti on. In addition to availab le nonlin ear Cosserat and micro morphi c theor ies, general formulations of elastoviscoplastic behaviour are proposed for microdilatation and microstretch continua. A microstrain theory is introduced that is based on six additional degrees of freedom describing the pure straining of the microstructural element. In each case, balance equations and boundary conditions are derived, decompositions of the finite strain measures into elastic and plastic parts are provided. The formulation of finite deformation elastoviscoplastic constitutive equations relies on the introduction of the free energy and dissipation potentials, thus complying with requirements of continuum ther- modynamics. Some guidelines for the sele ction of a suitable higher order model for a given material close the discu ssion . 2006 Elsevier Ltd. All rights reserved. Keywords: Continuum mechanics; Higher order continua; Micromorphic; Cosserat; Elastoviscoplasticity; Finite deformation 1. Introduction Incorporating some effects of characteristic lengths of the materials microstructure into constitutive mod- eling is possible via the mechanics of generalized continua. Generalized continua based on the assumption of local action, meaning that stress at a point depends only on quantities defined at that point, can be classified into two main groups ( Trostel, 1988; Forest, 2005). Higher grade continua are characterized by higher order spatial derivatives of the displacement field as in the second gradient theory of Mindlin ( Mindlin and Eshel, 1968; Germain, 1973a; Maugin, 1979), or by the gradient of internal variables as in strain gradient plasticity (Aifantis, 1987). Higher order continua are endowed with additional degrees of freedom that are a priori inde- pendent from the usual translational degrees of freedom, namely the three components of the displacement vector of the continuum point. The most recent and comprehensive account of the mechanics of higher order 0020-7683/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2006.05.012 * Corresponding author. Tel.: +33 1 60 76 30 51; fax: +33 1 60 76 31 50. E-mail addresses: samuel.forest@e nsmp.fr (S. Forest), [email protected] (R. Sievert). International Journal of Solids and Structures 43 (2006) 7224–7245 www.elsevier.com/locate/ijsolstr

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